- Commit
- 45e040d1a001023d7831dd72a1dc8acc2c579511
- Parent
- 841a1a48a14c12e33c2306032057b424e564b5f9
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Clarified the PBW theorem
Added a remark on the fact that the basis we work with are all ordered
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Clarified the PBW theorem
Added a remark on the fact that the basis we work with are all ordered
1 file changed, 4 insertions, 3 deletions
Status | File Name | N° Changes | Insertions | Deletions |
Modified | sections/introduction.tex | 7 | 4 | 3 |
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -655,9 +655,10 @@ we find\dots \begin{theorem}[Poincaré-Birkoff-Witt] Let \(\mathfrak{g}\) be a Lie algebra over \(K\) and \(\{X_i\}_i \subset - \mathfrak{g}\) be a basis for \(\mathfrak{g}\). Then \(\{X_{i_1} \cdot - X_{i_2} \cdots X_{i_n} : n \ge 0, i_1 \le i_2 \le \cdots \le i_n\}\) is a - basis for \(\mathcal{U}(\mathfrak{g})\). + \mathfrak{g}\) be an orderer basis for \(\mathfrak{g}\) -- i.e. a basis + indexed by an ordered set. Then \(\{X_{i_1} \cdot X_{i_2} \cdots X_{i_n} : n + \ge 0, i_1 \le i_2 \le \cdots \le i_n\}\) is a basis for + \(\mathcal{U}(\mathfrak{g})\). \end{theorem} The Poincaré-Birkoff-Witt Theorem is hugely important and will come up again